Hamiltonian cycles in powers of infinite graphs

Status open high confidence

Both conjectures remain open for general countable graphs that are not locally finite. The square and cube cases are fully settled for locally finite graphs — the locally finite result was already credited to Georgakopoulos's 2007 Oberwolfach report at the time of posting, and the complete proof appeared in *Advances in Mathematics* in 2009 — but no verified paper has resolved either conjecture for countable graphs with vertices of unbounded degree.

Cited literature (2)

• partial Infinite Hamilton Cycles in Squares of Locally Finite Graphs (2009)
  Agelos Georgakopoulos · Advances in Mathematics

  Provides the full journal proof that the square of every 2-connected locally finite graph contains a Hamilton circle (extending Fleischner's theorem to locally finite graphs), formalising the result cited in the OPG problem statement as already known.

• partial Hamiltonicity in Locally Finite Graphs: Two Extensions and a Counterexample (2018)
  Karl Heuer · Electronic Journal of Combinatorics · arXiv:1701.06029

  Gives a sufficient condition for the square of a locally finite graph to contain a Hamilton circle and answers a related Mohar question about unique Hamilton circles, but remains within the locally finite setting and does not address the stated conjectures for general countable graphs.

Reviewer notes. The Mohar problem page (verified) explicitly states that both conjectures remain open for the non-locally-finite countable case. The 2009 Advances in Mathematics paper formalises the locally finite result already credited at posting time. The ScienceDirect page for the 2009 paper returned HTTP 403; the paper was instead confirmed via the author's Warwick faculty page. No paper found after 2007 addresses cubes or squares of general countable (possibly non-locally-finite) graphs.

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